ADDITIONAL EXERCISES FOR MATHEMATICS 10BFALL 2009Chapter 7The exercises are organized by sections of the course text (Colley, Vector Calculus, ThirdEdition).Exercises for Colley, Section 7.1S1. Find an equation defining the tangent plane to the parametrized surface x(u, v) =(u+v, u−v, v) at the point p = (1, 1, 0). [Hint: The first step is to find (u, v) such that x(u, v) = p).S2. Give a set of parametric equations for the surface of revolution obtained by revolvingthe graph of the function x = sin z (0 ≤ z ≤ π) about the z-axis.S3. Find the surface area for the piece of the unit sphere x2+ y2+ z2= 1 cut out by theconical region z ≥px2+ y2.Exercises for Colley, Section 7.2S1. Find the surface area for the portion of the graph of z = 10 + 2x − 3y over the squarewith vertices (0, 0), (2, 0), (0, 2), and (2, 2).S2. Find the surface area for the portion of the graph of z = 4 + x2− y2over the diskdefined by x2+ y2≤ 1.S3. Find the surface area for the portion of the graph of z = xy over the disk defined byx2+ y2≤ 16.S4. A thin conical shell is given in coordinates by z = 4 − 2px2+ y2, where 0 ≤ z ≤ 4,and the density at each point is proportional to the distance between the point and the z−axis(hence is kpx2+ y2for some constant k. Find the total mass of this shell.In each of exercises S5–S7 below, evaluate the surface integralR RSF· dS, for the given choicesof the vector field (= vector valued function) F and the oriented surface S. In each case take theupward pointing normal orientation for S.S5. F(x, y, z) = (3z, 4, y) and S is the portion of the plane x + y + z = 1 in the first octant.S6. F(x, y, z) = (x, y, z) and S is the portion of the paraboloid z = 9 − x2− y2lying inthe upper half-space defined by z ≥ 0.1S7. F(x, y, z) = (x, y, z) and S is the portion of the sphere x2+ y2+ z2= 1 in the firstoctant.Exercises for Colley, Section 7.3S1. Using Stokes’ Theorem, calculate the line integralZΓF · dxwhere F(x, y, z) = (2y, 3z, x) and Γ bounds the triangle with vertices (0, 0, 0), (0, 2, 0), (1, 1, 1).S2. Suppose that D is the solid triangle consisting of all points which are on the planewith equation 2x +2y + z = 6 and in the first octant, and let Γ be the boundary of D parametrizedin the counterclockwise sense. Evaluate the line integralZΓF · dxwhere F(x, y, z) = (−y2, z, x).S3. Evaluate the surface integralR RSF · dS, where F(x, y, z) = (x2z, −y, xyz) and S isthe boundary of the cube defined by the inequalities 0 ≤ x ≤ a, 0 ≤ y ≤ a, 0 ≤ z ≤ a; as usual,take the normal to the surface to point outwards.S4. Evaluate the surface integralR RSF · dS, where F(x, y, z) = (z + 2x, x3y, −2z) and Sis the boundary of the hemisphere defined by the inequalities x2+ y2+ z2≤ 1 and 0 ≤ z ≤ 1; asusual, take the normal to the surface to point outwards.S5. Evaluate the surface integralR RS(∇ × F) · dS, where S is the upper hemisphere of thesphere x2+ y2+ z2= 16, the vector field F(x, y, z) = (x2+ y − 4, 3xy, 2xz + z2), and the normalorientation of S is upward.S6. Suppose that f and g are functions with continuous partial derivatives (defined on agiven region). Prove that ∇ × (f ∇g) = (∇f) × (∇g).S7. Evaluate the surface integralR RSF · dS, where S is the unit sphere with the outwardnormal and F(x, y, z) = (x3, y3, z3).S8. Evaluate the surface integralR RSF · dS, wheere S bounds the cylinder defined byx2+ y2≤ 1 and 0 ≤ z ≤ 1, and F(x, y, z) = (1, 1, z(x2+ y2)2).S9. Let F be a vector field defined on all of 3-space such that ∇ · F = 0 and ∇ × F = 0.Prove that F = ∇g, where g satisfies ∇2g = 0 (in other words, g is harmonic).S10. Suppose that the vector field F is defined and has continuous partial derivatives on aregion containing the closed surface S and the region W bounded by S, and assume further that therestriction of F to S is tangent to S at all points of the latter. Prove thatR R RW(∇ · F) dV = 0.2Exercises for Colley, Section 7.4S1. Let F be a vector field defined in a region of coordinate 3-space. An integratingfactor for F is a positive valued function λ defined on the region such that λ has continuous partialderivatives and λF = ∇g for some function g. Prove that if F has an integrating factor, then Fand ∇ × F are perpendicular to each other. [Hint: What can we say about the curl of λF? Whyis the curl of F equal to ∇(logeλ) × F, and how can one derive the conclusion of the problem fromthis?S2. Let F be the vector field F(x, y, z) = (y, z, x). Show that there is no integrating factorfor F on the open first octant defined by x, y, z > 0.S3. Let F = (P, Q, R) be a vector field defined on a region of coordinate 3-space, anddefine the Laplacian ∇2F by the coordinate-wise formula (∇2P, ∇2Q, ∇2R). As in the case ofscalar functions, we shall say that F is harmonic if ∇2F = 0. Using Exercise 15(a), show that F isharmonic if both ∇ × F = 0 and ∇ · F =